*Corresponding author. Tel.:#1-713-743-3798; fax:#1-713-743-3798. E-mail address:[email protected] (P. Thompson).
24 (2000) 1285}1313
Learning from the experience of others:
Parameter uncertainty and economic growth
in a model of creative destruction
Peter Thompson
*
Department of Economics, University of Houston, Houston, TX 77204-5882, USA Received 1 April 1998; accepted 29 March 1999
Abstract
This paper analyzes a quality-ladder model of economic growth incorporating uncer-tainty about the e$ciency of R&D. A central premise of the paper is that designing appropriate technology policies is more di$cult when one is at the cutting edge of technology. In technological laggards, information gleaned from observations of more advanced countries provides noisy signals about the e!ort required to develop a speci"c product generation, and aids in the design of policy. The paper shows how these signals in#uence growth rates and technology policies. ( 2000 Elsevier Science B.V. All rights reserved.
JEL classixcation: O3
Keywords: Growth; Quality ladders; Information
1. Introduction
This paper analyzes a quality-ladder model of economic growth incorporat-ing uncertainty about parameters governincorporat-ing the e$ciency of R&D, and learning
about those parameters. A central premise of this paper is that designing appropriate technology policies is more di$cult when one is at the cutting edge of technology. In technological laggards, information gleaned from observations of more advanced countries provides noisy signals about the e!ort required to develop a speci"c product generation, and aids in the design of policy. Firms in less advanced countries are therefore guided more accurately to undertake research of appropriate intensity, and planners can set more precise policies. In short, there is a greater correlation in technological laggards between ex ante and ex post optimal behavior in both competitive and e$cient equilibria.
In order to highlight the e!ect of information on comparative growth, I construct a model which strips away &traditional' sources of international linkages that have been extensively analyzed elsewhere. Thus the model econ-omy contains no physical capital, no trade, no foreign direct investment, and no technology transfer in the traditional sense of imitation. What is left is a novel source of international growth linkages in which di!erences in average growth rates are driven only by the ability to observe parameters in more advanced countries. The model explicitly acknowledges the conventional wisdom that technology policies are somehow easier to design when one is not at the frontier, a feature of comparative growth that has not previously been explored.
I develop a version of the quality-ladder model due to Aghion and Howitt (1992), Grossman and Helpman (1991) and Segerstrom et al. (1990). The main extension in this paper is that the e$ciency of R&D in the development of each product generation depends on a random variable that is assumed to be correlated across countries, and in contrast to previous work I focus primarily on the non-stationary equilibria that arise among technological laggards. The learning mechanism by which technological laggards update their beliefs about parameters governing the e$ciency of R&D e!ort is very general and may be of independent interest. It allows for imperfectly observable R&D e$ciencies in more advanced countries as well as imperfect correlation of R&D e$ciencies across countries. Moreover, learning need not be optimal in any statistical sense, although two examples with Bayesian updating of prior beliefs are provided.
The main results of the paper concern the properties of the model in technolo-gical laggards that adopt optimal policies, and those that follow alaissez faire
signals are, the greater is the parameter space for which signals raise growth in the e$cient equilibrium.
The model is less informative about expected growth under laissez faire. I have not been able to establish analogous conditions under which signals raise the expected instantaneous growth rate, nor have I been able to generate a counterexample. Nonetheless, while the implications of signals for catch-up remain an open question underlaissez faire, there are some interesting"ndings. Prominent among these is a contrast between the e$cient and laissez faire
models in the e!ect that news about the e$ciency of R&D in the next race has on R&D e!ort in the current race. News suggesting that R&D will be parti-cularly e$cient in the next race encourages the social planner to raise R&D e!ort in the current race: the news makes it more desirable to complete the current race. In contrast, the same news lowers R&D e!ort in thelaissez faire
equilibrium, because it implies that the winner of the current race is likely to enjoy monopoly pro"ts for only a short period of time. That is, signals about the e$ciency of R&D in future generations that raise the expected intensity of R&D in the next race are good news from the perspective of the social planner, but bad news for"rms participating in the current R&D race.
The layout of the paper is as follows. Section 2 presents the model, The competitive equilibrium is characterized and contrasted with the solution to the social planner's problem. Section 3 describes how signals from more advanced countries inform technological laggards about the productivity of their own R&D programs. Section 4 presents the main results on com-parative growth, and Section 5 concludes. All proofs are provided in Appendix A.
2. The model
I employ a simple quality ladder framework, based on the seminal work of Aghion and Howitt (1992), Grossman and Helpman (1991), and Segerstrom et al. (1990). The world consists of a number of countries, each consisting of a single manufacturing sector and a research sector. The research sector in each country is characterized by a sequence of patent races, each aimed at improving the quality of the country's existing state-of-the-art product. The duration of a patent race depends in part on a random variable that governs the e$ciency of R&D, and whose value is not known with certainty during the race. The winner of a patent race employs limit pricing to secure a temporary monopoly that lasts until the next innovation. Only the highest-quality product developed within a country at any point in time is manufactured. There is no trade, and there are no transfers of technology across countries. That is, all knowledge required to manufacture a given product generation must be developed locally. The only sense, therefore, in which the model might be interpreted as a multi-country model is that countries may be able to learn something from other countries about the value of the R&D e$ciency parameter. The remainder of this sec-tion considers an arbitrary country currently engaged in developing product generationq.
2.1. Consumers
A representative consumer maximizes the expected present value of lifetime utility,
max EtP=
t
e~o(q~t)lnu(q) dq, (1)
subject to an intertemporal budget constraint,
1Throughout the paper, superscripts denote powers of variables and subscripts are used for indexing purposes.
endowed with ¸units of labor earning a wage rate of w(t). The consumption index satis"es1
u(t)"
C
q(t+)~1j/1
jjx
j(t)
D
, (3)wherex
j(t) denotes consumption of thejth generation of the good,j'1 is the
proportional improvement in quality between consecutive generations, and
q(t)!1 denotes the number of product improvements that have been made available in the country by timet.
Homothetic preferences ensure separability of expenditure and prices, and the consumer's problem can be solved in two stages. In the"rst stage, the consumer allocates expenditure, m(t) across products. It will be assumed below that marginal cost is equal tow(t) for all"rms, and that each generation of goods can only be produced by the single"rm that holds the relevant patent. The prefer-ences given by Eq. (3) imply that quantity and quality are perfect substitutes. The consumer therefore purchases only the single good with the lowest adjusted price. Following standard practice, I assume that when quality-adjusted prices are equal, the highest quality product is consumed. Bertrand competition in this setting induces the holder of the national state-of-the-art patent to set a limit price ofjw(t). Only the state-of-the-art product is consumed and the monopoly leader captures revenue of m(t). In the second stage, the consumer allocates expenditure over time, which satis"es the familiar Euler equation, m5(t)/m(t)"r(t)!o. I let expenditure be the numeraire so that
m(t)"1. It then follows that the market rate of interest,r(t) is always equal to the discount rate,o.
2.2. Firms
Manufacturing requires only labor,¸
x(t), and one unit is required to produce
one unit of output of any good. The owner of the state-of-the-art patent sets a markup ofjover marginal cost, which secures a temporary monopoly and yields instantaneous pro"ts n(t)"(j!1)m(t)/j"(j!1)w(t)¸
x(t). All "rms
owning patents on earlier product generations earn zero pro"ts. Asm(t)"1, it follows thatw(t)"1/j¸
x(t) andn(t)"(j!1)/j.
Denote withv
q(t) the discounted pro"ts of the successful innovator of theqth
generation product. R&D also requires only labor. Let¸
q(i,t) denote the labor "rm idevotes to the race for the qth patent, and let¸
2As dtis arbitrarily close to zero, a
q(t)¸q(t)bdtwill also be arbitrarily close to zero almost everywhere as long as E(a
q) is"nite and¸q(t)bis bounded. The latter is bounded by the resource constraint, while the former is assumed below.
3Note also that this formulation of the R&D technology implies that individual "rms face constant returns to scale in R&D, leaving their size indeterminate, but returns to scale in aggregate R&D are diminishing. The R&D technology allows us to restrict attention to a representative"rm, irrespective of aggregate returns to scale in R&D.
4Aghion and Howitt (1992) consider an example in whicha
qis random but its value is known during the race.
5The assumption of zero correlation across product generations is made for analytical conveni-ence. The key insights of the model are not sensitive to the introduction of serial correlation across product generations. What is central to the paper is the assumption that countries have something to learn abouta
qfrom other countries.
6More precisely, one should writeF
n(q,t)(aq) to re#ect the fact that the number of signals varies with calendar time and product generation. I shall use the shorthand notation to avoid unnecessary clutter.
aggregate labor devoted to the race. The expected discounted pro"ts of"rmiare
E[v
which equals zero under the assumption of free entry to the patent race. The terma
q¸q(t)bdt,b(1, is the probability2that the patent race will be won in the
next momentary interval dt, and¸
q(i,t)/¸q(t) is the probability that, if the race is
won, "rm i will be the winner. Innovations are Poisson events with a time-varying intensitya
q¸q(t)bdt, and the duration,q, of the race for product
genera-tionqis given by the time-varying exponential distribution expM:q0a
q¸q(t)bdtN,
wheret"0 denotes the time at which the race began. For any"xed¸
q,aqis
proportional to the arrival intensity, and may naturally be interpreted as a measure of R&D e$ciency.3
The R&D e$ciency parameter a
q is not known with certainty during the
patent race.4Before and during a race,"rms assign toa
qa subjective
distribu-tion F
n(aq). The main features of aq are: (i) it is independent across product
generations, and (ii) it is correlated across countries.5Thus, while a country at the technological frontier must make decisions based only on the prior distribu-tion of a
q, countries which have the opportunity to look ahead to other
countries'experiences with the same generation of technology will have a better idea about the value thata
qis likely to take. The subscriptndenotes the number
of observations available on a
q and at any point in time it may vary across
product generations.6These characteristics of F
nwill be explored in detail in
Assumption 2.1. LetF(a),F
0(a)denote the prior distribution of a, letE(a)(R denote its prior mean, and let¹denote the product generation being developed at the world technological frontier. (i) F(0)"0, (ii) F(a) is diwerentiable, and (iii)
F
0(aT`j)"F(a),∀j50.
Assumption 2.1 restrictsato the positive half line (thereby ensuring that no country enters a no-growth trap), and states that no observations are available for any product generation not yet developed somewhere.
The memoryless property of the R&D production function implies that innovations are independent Poisson events. As long as no new information about the value of a
q is received, the intensity of R&D is constant for the
duration of any race and the time to the next innovation is exponentially distributed. Note also thatv
q(t)}the value of winning theqth race}depends on
the duration of the (q#1)th race. Thus, conditional ona
q`1, the expected value
of winning theqth race can be written as
E
Integrating Eq. (5) over all possible values of a
q`1 yields the unconditional
expected value of winning theqth race, E
n(vq(t)). Then, combining with the zero
pro"t condition Eq. (4), the equilibrium intensity of aggregate R&D satis"es
¸
The intensity of R&D in the race for theqth monopoly depends not only on the expected value of a
q, but also on the outcome of the race for the next
generation. The terma
q`1¸q`1(t)bdtdenotes the probability that the race for
the (q#1)th monopoly ends in the interval dt, where ¸
q`1(t) denotes the
intensity of R&D in the (q#1)th race that is believed to be optimal given the information available at timet;F
n(aq`1) denotes the current subjective
distribu-tion for a
q`1. The instantaneous probability that the qth monopoly ends its
tenure enters as an addition to the rate of discount. The expected payo!to the winner of the qth race is therefore equivalent to an instantaneous pro"t#ow, (j!1)/j, earned in perpetuity with an interest rate ofo#a
q`1¸q`1(t)b.
2.3. The competitive equilibrium
Substituting for w(t) in Eq. (6) and using the full-employment constraint,
¸
generationqis de"ned by the"xed point expression,
¸
q(t)1~b"(j!1)(¸!¸q(t))En(aq)
P
=0
dF
n(aq`1)
o#a
q`1¸q`1(t)b
. (7)
For technological laggards, laissez faire equilibria must satisfy the non-linear"rst-order di!erence equation written in implicit form in Eq. (7). Bound-ary conditions are given by equilibria that apply at the world technological frontier.
The"rst lemma establishes that there is a unique stationary equilibrium at the frontier.
Lemma 2.2. A country at the technological frontier has a unique stationary com-petitive equilibrium with positive R&D intensity, ¸cT, satisfying
(¸cT)1~b
¸!¸cT"(j!1)E(a)
P
=0
dF(a)
(o#a(¸cT)b). (8)
Periodic equilibria at the world technological frontier cannot always be ruled out in this model. One can prove their existence by following the analysis in Aghion and Howitt (1992), (Section 3A). However, I will follow Aghion and Howitt (1992), and others, in restricting attention to the unique stationary equilibrium at the world technological frontier.
I am now in a position to provide conditions for the existence of a unique competitive equilibrium during the race for product generation q(¹ in a technological laggard:
Theorem 2.3. Assume there is a unique sequence,F
n(aq),Fn(aq`1),2,Fn(aT~1), of
subjective distributions with positive support. Then, there is a unique competitive equilibrium intensity of R&D,¸cq, during the race for generationq(¹.
Theorem 2.3 states that if the intensity of R&D at the world technological frontier is uniquely de"ned then so is the intensity of R&D during the race for the current product generation in a technological laggard. The intuition behind the absence of multiple, periodic equilibria in laggards is straightforward. The possibility of cyclical equilibria at the frontier arises because there is no bound-ary condition to pin down the value of¸
T`jfor anyj'0. In contrast, restricting
2.4. The social planner
This section characterizes the e$cient intensities of R&D. It is well known that laissez faire equilibria in this class of models are not typically e$cient. Aghion and Howitt (1992), Dinopoulos (1994), and Grossman and Helpman (1991), among others, discuss the opposing externalities that lead to an ambigu-ous relationship between competitive and e$cient R&D intensities. In this and the following subsections, the externalities that have been identi"ed previously in steady-state analysis will be derived for technological laggards. In addition, however, there are some important di!erences between the competitive and e$cient intensities of R&D that do not arise when attention is restricted to countries at the technological frontier.
Indirect utility in this model is given by
C(t)"
P
=t
e~o(q~t)[(q(q)!1) lnj#ln(¸!¸
q(q))] dq, (9)
whereq(t) denotes the product generation for which researchers are racing at timet. Thus, the consumption good at timetis indexed byq(t)!1.
The social planner maximizes the expected value of Eq. (9) subject to the technological conditions governing the rates of innovation;q(t) is an integer-valued step function where the steps are formed from a Poisson process with magnitude one and intensitya
q¸q(t)b. The Bellman equation for this problem is
max
Lq(t) o<
q(t)"(q!1) lnj#ln(¸!¸q(t))
#[<
q`1(t)!<q(t)]
P
=
0 a
q¸q(t)bdFn(aq), (10)
where<
q(t) denotes the value function at timetwhen researchers are racing for
product generationq. The right-hand side of the Bellman equation consists of three terms. The "rst two terms describe the #ow of consumption bene"ts received during the race for product generationq. The third term is the jump in the value function that occurs when generationqis developed, multiplied by the expected probability that the innovation takes place in the next instant. The maximized sum of these three terms equals the interesto<
q(t) that can be earned
on a risk-free bond of size<
q(t).
Di!erentiating Eq. (10) yields the"rst-order condition
¸H
q(t)1~b"bEn(aq)(¸!¸Hq(t))[<q`1(t)!<q(t)]. (11)
Although the social planner's problem appears simple, its solution is not easily obtained. In general, the subjective distributions,F
n(aq), are not stationary over
7The comparative statics in Lemma 2.4 are familiar from analogous results in Aghion and Howitt (1992) and Grossman and Helpman (1991).
8The clearest prior statement of this result is given by Grossman and Helpman (1991), (pp. 104, 105) for the case whereb"1 andais known.
stationary problems apply only for countries at the technological frontier. Following the analysis of Section 2.3, I"rst characterize the solution that applies at the frontier (Lemma 2.4). E$cient and laissez faire rates of growth at the frontier are compared in Proposition 2.5. Existence and uniqueness of the solution to the planner's problem is then established for technological laggards by backward induction (Theorem 2.7). In the following subsection, I compare the e$cient andlaissez faire R&D intensities for technological laggards, and characterize the optimal subsidy to R&D (Proposition 2.8).
Recall that¹is the lowest index of undiscovered technologies in the world. For a country attempting to develop generation T the solution to Eq. (9) takes the form o<
T"A#(¹!1) lnj, where A is a coe$cient to be determined.
Thus,<
T`1!<T"o~1lnjand¸HTsatis"es the"xed-point expression
¸H1~b
T "
blnjE(a)(¸!¸H
T)
o . (12)
It is easy to see that a unique interior solution to Eq. (12) exists, in which case
<
Tsatis"es
o<
T"(¹!1) lnj#ln(¸!¸HT)#
E(a)¸H
Tlnj
o , (13)
where¸H
Tis the increasing function of E(a) de"ned in Eq. (12). Lemma 2.4 follows
immediately.7
Lemma 2.4. At the technological frontier, there is a unique solution to the social planner's problem, with R&D intensity increasing inb,jandE(a), and decreasing ino.
Let EgcT denote the expected growth rate of quality at the frontier under
laissez faireand letEgH
Tdenote the corresponding expected growth rate in the
e$cient equilibrium. The following proposition is readily established.8
Proposition 2.5. If ¸ is suzciently large, there exists a pair of values,
1(j
0(j0(R, such that iwj3[j0,j0], thenEgHT'EgcT;j0[j0]is decreasing
Proposition 2.5 highlights the fact that economy size is critical in determining whether or not the competitive equilibrium generates a faster growth rate than is socially optimal. Jones and Williams (1997) have calibrated a simple R&D-based growth model to US data. They concluded that the optimal intensity of R&D may be as much as four times larger than the current intensity. Following their lead, it will be assumed throughout the remainder of this paper that
j0(j(j0.
Assumption 2.6. EgH
T'EgcT.
I turn now to the social planner's problem for technological laggards. First, it can be con"rmed that a unique solution exists:
Theorem 2.7. Consider a country currently racing for product generationq(¹, that has posterior means E
n(aq), En(aq`1),2, En(aT~1) of the R&D productivity parameters. Then,(i)there is a unique solution to the social planner's problem that depends on the sequence MME
n(aj)NjT/~1q, E(a)N; (ii) R&D intensity is increasing in each element of the sequence.
Theorem 2.7 highlights the forward-looking nature of the social planner in technological laggards. If the planner expects R&D e!ort in future generations to be especially productive, then he will accelerate the current R&D race in order to bring forward the expected arrival date of those future generations. Conversely, a pessimistic outlook for future R&D races retards the current growth race. Even though technological leapfrogging is excluded from this model, the planner's current R&D policies always depend on all technolo-gies that have been developed elsewhere but that remain to be developed at home.
Note also that the solution to the social planner's problem depends only on the sequence of subjective means, while the competitive equilibrium intensity of R&D depends on all moments of the subjective distribution F
n(aq`1). The
intuition behind this result is straightfoward. Firms engaging in the race to generate product generation q care about the expected present value of the monopoly pro"ts that accrue to the winner. As the expected present value of monopoly pro"ts is a nonlinear function ofa
q, all moments ofFn(aq`1) enter into
the solution. In contrast, the social bene"ts of any innovation planner last inde"nitely, so that the duration of the monopoly for product generationqis not of direct concern to the social planner.
2.5. Comparing ezcient and competitive equilibria in technological laggards
9Use Eq. (11) in Eq. (10) to remove<
q(t). This yields an expression relating¸Hq to<q`1. Then
update (11) and (10) by one product generation and combine them to remove<
q`2!<q`1. This
generates an expression relating¸H
q`1to<q`1. The two new expressions so obtained can then be
combined to eliminate< q`1.
10The substitution of lnj for (j!1) confounds two market failures. The"rst is the proxt destructione!ect: a new monopoly earns pro"ts, (j!1), only by destroying the pro"ts earned by the previous monopoly. As the social planner is not concerned with the identity of the"rm that currently earns pro"ts, this private value of innovation is ignored. The second is theconsumer apppropriability e!ect: a social planner values the increment to consumer surplus, lnj, but this value is not appropriable by"rms who therefore ignore it.
writing the optimal intensity of R&D,¸H
q(t) as a function9of¸Hq`1(t),
and which can be compared with thelaissez faireequilibrium,
¸cq(t)1~b
Note that whenq"¹, the second term on the RHS of Eq. (14) vanishes. The di!erences between the "rst terms on the RHS of Eqs. (14) and (15) re#ect market failures that are now familiar in quality ladder models. First, the solution to the planner's problem substitutes lnjfor (j!1) because the planner cares about consumer surplus while what matters in the competitive equilibrium is
"rm pro"ts.10Second, the planner's solution includesb, which is absent from the competitive equilibrium. This di!erence is acongestion externality that arises because the planner recognizes that each"rm contributes to aggregate diminish-ing returns to scale in R&D, while returns to scale are constant for the individual
"rm. Third, the planner discounts the future at the rate o, while the "rm discounts the pro"t #ow at the rate o#a
q`1¸cq`1(t)b. This di!erence arises
because"rms survive only to the next innovation while the social value of an innovation lasts forever. These features of the model are well known from steady-state analyses.
Among technological laggards, however, there is a fourth divergence between the e$cient and competitive outcomes. The planner's problem includes a term that depends on the di!erence between the optimal intensity of R&D in the current race and the intensity of R&D that is currently expected to be optimal in the next race. This additional term in Eq. (14) re#ects the forward-looking nature of the social planner:¸H
q`1(t) is in fact a summary statistic for the current
A subsidy, s
q, to R&D can be employed to equate ¸Hq and ¸cq. Naturally, s
qincreases with the distance between the e$cient and competitive intensities of
R&D. At the technological frontier,¸H
Tand¸cTare constant, and so it follows
that the optimal subsidy to R&D,sH
T, is also constant. Substituting for the unit
cost of R&D in Eq. (6) and then comparing Eq. (8) with Eq. (12), the optimal subsidy at the frontier is given by
sH
which, by Assumption 2.6, is positive. In contrast, for q(¹,sH
q is a random variable that varies across product
generations, and within a product generation whenever a new signal is received. Moreover, one cannot even signsH
q; bad news about the prospects for developing
product generation q#1 increases ¸cq but reduces ¸H
q, and in the face of
su$ciently bad news the former may be larger.
Proposition 2.8. The optimal subsidy, sH
q,q(¹, for technological laggards is a random variable, not necessarily with strictly positive support. The optimal subsidy at the world technological frontier, sH
T, is constant and positive across product generations and countries.
3. International signals of R&D e7ciency
It is necessary at this stage to impose some minimal structure on the relation-ship between the prior distribution, F(a), a technologically lagging country's posterior distribution ofa
q,Fn(aq), and thenrealizations ofaqthat a country has
observed. In this section I make two assumptions that are su$cient to enable me to say something useful in Section 4 about comparative growth.
The "rst assumption is that the signals received and the rules used to transform the signals into an expectation generate values of E
n(aq) that are
unconditionally unbiased.
Assumption 3.1. LetH(E
n(aq)Daq)denote the conditional distribution ofEn(aq)when the unobserved ezciency of R&D isa
q, and letE(En(aqDaq))":En(aq) dHdenote its conditional expectation. Then:E(E
n(aq)Daq) dF"E(a).
The second requirement is that observations of the values of a
q realized in
more advanced countries provide useful information about the value ofa q. That
is, large values of E
n(aq) should be more likely when the true (unobserved) value
of a
q is large. This requirement is made precise in the sense of "rst-order
Assumption 3.2. For any aA
q'a@q,H(En(aq)DaqA)4H(En(aq)Da@q). If this holds as a strict inequality for someE
n(aq)'0, then the signals are informative.
These two assumptions can accommodate a large variety of stochastic envi-ronments. At one extreme, the parameter a
q is perfectly observed after the
completion of an innovation race, but it is imperfectly correlated across coun-tries. By observing realizations of a
q in advanced countries, a technological
laggard can learn something about the distribution from which its own value of
a
qwill be drawn. At the other extreme,aqis identical across countries but it is
imperfectly observed. For example, a laggard may observe the time or cost required to produce an innovation, and from this it can infer something about
a
q. Assumptions 3.1 and 3.2 can accommodate these extremes as well as a
combi-nation of imperfect correlation and imperfect signals.
The two assumptions also do not require that signal processing be optimal. However, two parametric examples in which the updating rule is Bayesian are provided here.
Example 3.3 (Perfect information, imperfect correlation). Assume that the country-speci"c e$ciency parameter,a
qis a random variable drawn from an
exponential distribution with unknown parameter v
q. The parameter of the
distribution is speci"c to the product generation, but not to the country. Assume further thatv
qis itself a random variable that has a prior gamma distribution
with parametersa"2 andb'0. The prior density forais given by
f(a)"
P
=Assume now that a country has observednrealizations ofa
qwith meank, and
that prior beliefs are updated by Bayes' rule. The 2-tupleMk,nNis a su$cient statistic for a
q and the posterior density function, derived in Appendix B, is
given by
f
n(aqDk)"
(n#2)(b#nk)n`2
(a#b#nk)n`3 . (18)
Of course, Eq. (18) contains the prior density as the special case in whichn"0. It is shown in Appendix B that Eq. (18) satis"es Assumptions 3.1 and 3.2.
Example 3.4 (Imperfect information, perfect correlation). Suppose that a qis the
same for all countries. The technological laggard observes the durations
t
1,t2,2,tn, of patent races inncountries at the technological frontier. Given the
technology of innovation described in Section 2, the durations are exponentially distributed with unknown parametera
11Taking limits of Eq. (18) one obtainsf"ea@k/k, and by the law of large numberskconverges on to 1/v
q.
all countries at the frontier, so one can choose units such thatRb"1. Assume further that the prior distribution of a
q is gamma with parametersa'0 and b'0. Then the posterior distribution ofa
qis gamma with parametersa#nand b#+ni
/1ti. This result is standard (e.g. De Groot, 1970), and is not analyzed in
the appendix.
While both examples are special cases of Assumptions 3.1 and 3.2, there is an important di!erence between them. In Example 3.4 a su$cient number of observations allows a country to know the value ofa
qprecisely. In Example 3.3,
in contrast, the technological laggard can never know its value ofa
qprecisely. As nPR, the limiting posterior distribution in Example 3.3 does not become degenerate, but rather converges with probability one to an exponential distri-bution with known parameterv
q.11
4. Comparative growth
This section focuses on the impact of signals on expected instantaneous growth rates. The main results are as follows. Growth rates for e$cient techno-logical laggards observing signals are more variable than they are for countries at the frontier. Signals do not always increase the expected growth rates of e$cient laggards, but they are more likely to do so when aggregate returns to scale in R&D do not diminish too rapidly and the equilibrium level of R&D employment is a small fraction of the labor force. Even under these conditions, I cannot establish that signals raise the average instantaneous growth rate of technological laggards adopting a policy oflaissez faire. There is an intuitive reason, explained below, why one might not expect alaissez faireequilibrium to exhibit a clear comparative growth result. However, this observation should be viewed with caution. I have also been unable to "nd an empirical counter-example in which the expected growth rate of a technological laggard is lower underlaissez faire than at the frontier, and so the e!ect of signals on growth underlaissez faireremains an open question.
signals. The answers to the two questions need not be the same in a stochastic environment, because the relationship between the two measures of growth is non-linear. Too see this, consider a country that lags one generation behind the frontier and observes precisely either of two e$ciency parameters, a
0and a1,
wherea
1'a0and each of which could have occurred with probability one half.
Let (j!1)a
0¸b0and (j!1)a1¸b1denote the growth rates that are realized when
each parameter is observed, and note that¸
1'¸0. The unconditional expected
growth rate is then 0.5(j!1)[a
0¸b0#a1¸b1]. The unconditional expected
dura-tion of the patent race is given by 0.5[(a
0¸b0)~1#(a1¸b1)~1]. Let¸Tdenote the
R&D intensity at the frontier. Then the signals raise the expected instantaneous growth rate only if (a0¸b
0#a1¸b1)/(a0#a1)'¸bTand they reduce the expected
duration of the patent race only if (a
0¸b0#a1¸b1)/(a0#a1)((¸0¸1/¸T)b. Given
parameter valuesb"1,a
0"1,a1"2,¸0"1 and¸1"2, the"rst inequality
requires that¸
T(1.6, while the second inequality requires that¸T(1.8. There
is a window in which it is possible that signals increase the expected instan-taneous growth rate while raising the expected duration of the patent race.
4.1. Ezcient technological laggards
The analysis of this section begins with a useful lemma.
Lemma 4.1. For any q(¹, E(a
q¸Hq(t)b)5E(a) E(¸Hq(t)b) with a strict inequality
[equality]if signals are informative[uninformative].
E$cient laggards will tend to raise their R&D intensity when the (unobser-ved) value of the R&D e$ciency parameter is high, and they will tend to reduce intensity when e$ciency is low. As all countries face the same unconditional distribution for the e$ciency parameter, the positive correlation betweena
qand
¸H
q(t)bimmediately yields the following result:
Proposition 4.2. Growth rates are more variable across product generations in ezcient laggards than they are in countries at the technological frontier.
At the technological frontier, a social planner chooses a constant intensity of R&D,¸H
T, and so the expected growth rate is given by EgHT"(j!1)E(a)¸HTb.
The social planner in a technological laggard, in contrast, chooses R&D in response to signals received, and the expected growth rate in this case is EgH
q(t)"(j!1)::a¸qH(t)bdHdF"(j!1)E(aq¸Hq(t)b)*(j!1)E(a)E(¸Hq(t)b).
Thus E(¸H
q(t)b)*¸HTbis a su$cient condition for technological laggards to grow
more rapidly on average than countries at the frontier. The di$culty in estab-lishing this inequality is that¸H
q(t)bdepends (non-randomly) on En(aq), which is
a random variable that in turn depends on the unobserved value of a
q.
Moreover, ¸H
12The latest available percentages from the OECD are: France, 2.0%; Germany 2.7%; Japan, 2.8%; United States, 1.9%.
13Bound et al. (1984) have suggested that returns to scale in R&D are approximately constant up to$100 million of expenditure, with decreasing returns setting in thereafter. Thompson (1996) exploited the relationship between equity price and R&D to obtain estimates ofbat the two-digit level ranging from 0.53 to 1.28, with a mean of 0.84.
Whether one can rank E(¸H
q(t)b) and¸HTbturns in large part on whether one can
show that ¸H
q(t)b is a convex function of En(aq) and of ¸Hq`1(t). The required
convexities do not always hold. However, the following lemma provides condi-tions under which they do.
Lemma 4.3. Ifb3(1/2, 1), there exists ane3(0, 1)such that for all ¸H
q(t)3(0,e¸)
and¸H
q`1(t)3(0,e¸),¸Hq(t)b, is a locally convex function ofEn(aq)and of¸Hq`1(t).
The lemma requires that aggregate returns to scale in R&D do not diminish too rapidly, and that in equilibrium R&D labor is a small enough fraction of the total labor force. Although one must be careful in comparing empirical assump-tions of an abstract model with data, the condiassump-tions seem plausible. First, R&D expenditures among even the most R&D-intensive countries are less than three percent of GDP.12Second, industry evidence suggests only weakly diminishing returns to scale.13Note also that the conditions of Lemma 4.3 are su$cient but not necessary. They become necessary conditions only in the limiting case that the signals to which the social planner is responding turn out to be completely uninformative.
Proposition 4.4. If the conditions of Lemma 4.3 hold, then for any q(¹, the following expected growth rates can be ranked:EgH
q'EgHT'EgcT.
A numerical example may provide some more intuition about these results. Assume that a
q may take either of two values, HIGH or LOW, with equal
probability. The social planner observes a signal about the e$ciency of R&D for each product generation behind the world technological frontier, and assumes that the signal is a precise predictor of its own country's R&D e$ciency. Table 1 reports the social planner's choice of R&D intensity,¸H
q, for each possible set
of realized signals. The conditional expected growth rates,gH
q, and the
uncondi-tional expected growth rates, EgH
q, are given for four di!erent degrees of signal
accuracy. Note that the choice of R&D e!ort depends only on the signals received. The expected growth rates, in contrast, also depend on the accuracy of those signals.
There are two channels through which signals increase the unconditional expected growth rate. The"rst is that signals direct the planner to devote more e!ort to R&D when signals suggest that it will be particularly e!ective. The second channel results from the convexity of the function¸Hqb(E
n(aq)) established
in Lemma 4.3. In the case where signals are completely uninformative, the"rst channel does nothing to raise the expected growth rate. Hence, Table 1 provides a useful decomposition of the sources of enhanced growth. At the frontier, the unconditional expected growth rate is 2.2%. When signals are completely uninformative, a social planner developing product generation¹!1 attains an expected growth rate of 3.1%, while for generation¹!2 it is 3.5%. Hence, the growth rate rises by 0.9% and 1.3%, respectively, simply as a result of the convexity. Informative signals raise the expected growth rate further. For example, on moving from uninformative to perfectly informative signals the unconditional expected growth rates increases from 3.1% to 4.2% in generation
¹!1, and from 3.5% to 4.6% in generation¹!2.
Of course, these observations do not imply that a social planner should randomly alter the R&D intensity to raise the expected growth rate. If signals are uninformative and the social planner knows this, the optimal policy is a constant intensity of R&D equal to the rate chosen at the world technological frontier. In contrast, when signals are precise the welfare-maximizing policy is to choose the R&D intensities indicated in Table 1. In this example, the planner alters R&D intensities under the possibly mistaken belief that the signals are perfectly informative. The welfare e!ect of signals therefore depends on the correspondence between the accuracy of signals and the social planner's evalu-ation of their accuracy. To say more, however, would require making further assumptions about the properties ofH(E
n(aq)Daq).
4.2. Technological laggards
Table 1
Numerical example: Social planners's problem
Signal observed Expected growth rate by signal accuracy! Product
generation a
T~2 aT~1
R&D e!ort
0.50" 0.60" 0.75" 1.00"
¹ } } 1.52 0.022 0.022 0.022 0.022
¹!1 } HIGH 4.83 0.062 0.066 0.072 0.083
} LOW 0.05 0.001 0.001 0.001 0.001
MEANS: 2.44 0.031 0.034 0.037 0.042
¹!2 HIGH HIGH 6.58 0.082 0.087 0.095 0.109
LOW HIGH 0.15 0.003 0.003 0.002 0.002
HIGH LOW 4.20 0.055 0.058 0.064 0.073
LOW LOW 0.03 0.001 0.001 0.005 0.000
MEANS: 2.74 0.035 0.037 0.041 0.046
Parameter values used in the example are:¸"100,b"0.7,o"0.05,j"1.04. The R&D e$ciency parameters area"M0.01, 0.02Nwith probabilitiesM0.5, 0.5N.
!Numbers in bold type are the unconditional expected growth rates; the remaining numbers are the expected growth rates conditional on each realization of the signals.
"Signal accuracy is de"ned as follows. The number refers to the probability thata
qisHIGH[LOW] when the signal observed isHIGH[LOW]. Thus, 0.50 in the"rst column de"nes a completely uninformative signal. The second and third columns indicate the signals are informative by the de"nition of Assumption 3.2, but are not precise. The fourth column indicates precise signals.
about the e$ciency of R&D in the current race induce more research when R&D is believed to be e!ective and less when it is believed to be relatively ine!ective. Under the same conditions as laid out in lemma 4.3, this e!ect promotes growth over the long run. On the other hand, if signals raise the average expected growth rate in, say, generation¹!1, the expected duration of the monopoly attained by the winner of the race for product generation¹!2 may be reduced. This channel will have a negative e!ect on R&D e!ort in the race for generation¹!2. Thus, while it is easy to show that, if the conditions of Lemma 4.3 apply, EgcT
~1'EgcT, I cannot show that EgcT~2'EgcT. However,
I have not been able to produce a counterexample in which EgcT
~2(EgcT. Thus,
the e!ect of signals on thelaissez fairegrowth rate in countries lagging the world technological frontier by at least two generations remains an open question.
Table 2, continuing the earlier example, provides corresponding data for the
Table 2
Numerical example:laissez faireequilibrium!
Product generation
Signal observed Expected growth rate by signal accuracy
a
T~2 aT~1 R&D e!ort 0.50 0.60 0.75 1.00
¹ } } 0.75 0.015 0.015 0.015 0.015
¹!1 } HIGH 1.77 0.025 0.027 0.029 0.033
LOW 0.08H 0.002 0.001 0.001 0.001
MEANS: 0.93 0.013 0.014 0.015 0.017
¹!2 HIGH HIGH 0.61 0.009 0.010 0.011 0.013
LOW HIGH 0.00 0.000 0.000 0.000 0.000
HIGH LOW 15.90H 0.181 0.193 0.211 0.241
LOW LOW 0.09H 0.002 0.001 0.005 0.000
MEANS: 4.15 0.048 0.051 0.056 0.064
!See notes to Table 1. Asterisks denote realizations that lead to alaissez faireintensity of R&D that exceeds the social planner's choice of R&D.
a high conditional growth expected rate in the race for product generation
¹!1 leads to a low expected growth rate in the race for generation¹!2. This re#ects, of course, the fact that observinga
T~1"HIGH is good news for the
social planner developing generation¹!2, but bad news for"rms engaged in the race for¹!2. Third, while thelaissez faireintensity of R&D at the frontier is generally less than the e$cient intensity of R&D (calling for a subsidy to R&D), there are three instances in which the social planner in a technological laggard would choose to tax R&D. These three instances correspond to the occasions on which the signal for a
T~1 is LOW. In production generation ¹!2, the prospect of a long tenure for the winner of the current R&D race encourages"rms to conduct more R&D inlaissez fairethan is socially desirable.
5. Conclusions
correlated with the R&D e$ciency in any country that develops the same product generation at a later date. My aim in this paper was to explore the implications of this information advantage for economic growth.
The principal conclusion is that the e!ects of information depend on whether technological laggards adopt e$cient technology policies, or whether they take an approach of laissez faire. When policies are optimally chosen, the model suggests that catch-up to the world technological frontier is likely. Su$cient conditions for catch-up are that aggregate returns to scale in R&D do not diminish too rapidly and that parameters yield equilibria in which R&D e!ort constitutes a small fraction of the total labor force. The parameter space for which catch-up takes place is expanded as signals become more informative. There are now numerous models in which technological catch-up is possible within a framework of endogenous growth. Most of these (e.g. Eaton and Kortum, 1997; Barro and Sala-i-Martin, 1995) rely on explicit forms of techno-logy transfer or imitation. This paper describes an alternative mechanism by which catch-up may be possible.
The results in this paper contrast with those obtained by Hopenhayn and Muniagurria (1996). They studied a one-sector endogenous growth model in which the subsidy to investment may be either positive or zero. They found that more frequent regime switches were likely to reduce growth and raise welfare. In the present paper, in which regime switching is information driven, the growth rate is likely to increase as a result of policy changes, although welfare need not be enhanced.
These results were obtained with a minimal set of assumptions on the transmission and utilization of information that includes, but is not restricted to, optimal updating of prior beliefs by Bayes'rule. I have not been able to replicate these results for thelaissez faireequilibrium. However, the analysis does reveal that, even though the optimal policy at the world technological frontier is always a subsidy to R&D, there are instances in which technological laggards would optimally choose a tax. Barro and Sala-i-Martin (1995) observed that we do not have any theory about convergence or divergence of government policies across countries. In a limited sense, this paper provides one. While the paper o!ers no insight as to why some countries might choose e$cient policies when others do not, it does indicate that, among countries that choose e$ -cient policies, the nature of the optimal intervention depends on one's technolo-gical ranking. As countries adopting e$cient policies catch up to the world technological frontier, their policies begin to look like those in other advanced countries.
are some simplifying assumptions that might matter in substantial ways. Consider, for example, my choice to exclude international trade. Given the market structure I have employed, the introduction of international trade to the model would have important consequences. Under Bertrand competition, qual-ity leaders drive out producers of inferior goods so that in a closed economy only the state of the art product is consumed. One might, therefore, worry about how the model would behave when di!erent qualities of goods coexist in a world with international trade. Of course, balanced trade constraints imply that wages will depend on the highest quality good that each country is capable of producing, so that it is not at all obvious that a quality leader in one country could drive out producers of inferior goods in other countries. However, this analysis remains to be done and, in this sense, the analysis in the present paper might best be interpreted as exploratory.
Acknowledgements
I am extremely grateful for extensive comments from"ve anonymous referees, and helpful suggestions on this and earlier drafts from Margaret Byrne, Michael Palumbo, and Costas Syropoulos.
Appendix A. Proofs
Proof of Lemma 2.2. By Assumption 2.1,F
n(aT`j)"F(a) and En(aT`j)"E(a),
∀j50. Hence if a stationary equilibrium with positive R&D e!ort exists, it must satisfy Eq. (8). Existence and uniqueness is established as follows. The LHS of Eq. (8) is monotonically increasing in ¸
T and continuous along the interval
¸
T3[0,¸); the LHS equals zero when ¸T"0 and approaches #R when
¸
TP¸. For E(a)3[0,R) the RHS of Eq. (8) is continuous, monotonically
decreasing in ¸
T, and equal to (j!1)E(a)/o when ¸T"0. Thus a unique
interior stationary equilibrium exists.
Lemma A.1. Assume F
n(aq`1)is uniquely dexned with positive support, and that
0(E
n(aq)(R. For any xxed ¸q`1'0, the mapping ¸q"G(¸q) dexned by Eq. (7) has a uniquexxed point¸cq3(0,¸).
Proof. G(¸
q)can be written as G(¸
q)"[(j!1)En(aq)]1@(1~b)/(¸q`1)(¸!¸q)1@(1~b),
where /(¸
q`1)"[:(o#aq`1¸bq`1)~1dFn(aq`1)]1@(1~b). If Fn(aq`1) is uniquely
de"ned and has positive support, then / is a well-de"ned function mapping [0,¸] into the closed interval [c,o~1] for some 0(c(o~1. Asb(1,G(¸
Fig. 1.
continuous and decreasing along the closed interval [0,¸]. Moreover,
G(0)"/(¸
q`1)[(j!1)En(aq)¸]1@(1~b)'0, while G(¸)"0. Hence, there is
a unique"xed point¸cq"G(¸cq) satisfying 0(¸cq(¸.
Proof of Theorem 2.3. This is straightforward with the use of Lemma A.1. Lemma 2.2 states that a unique ¸cT'0 exists. Then Lemma A.1 states that a unique ¸cT
~1'0 exists if Fn(aT) is uniquely de"ned with positive support.
Theorem 2.3 then follows by backward induction to product generationq.
Proof of Proposition 2.5. Note that¸1~b
T /(¸!¸T) is increasing in¸T.
Combin-ing Eqs. (8) and (12), yields
¸H
T'¸cT i!
1
:(o#a(¸c
T)b)~1dF(a)
'o(j!1)
blnj . (A.1)
Let /(¸,j) denote the expression on the left hand side of the inequality in Eq. (A.1). From Eq. (8) it is easy to see that:¸cTis increasing injand¸; for"xed
¸,/(¸,j) is bounded; and/(¸, 1)"o. Lett(j) denote the right-hand side of the inequality, and note that: t(1)"o/b;t@(j)'0; and lim
j?=t(j)"R. These
two functions are plotted in Fig. 1. Clearly, for values of jsmall enough and large enough, /(¸,j)(t(j). Furthermore, if ¸ is su$cently small, then
/(¸,j)(t(j) for all j. But /(¸,j) is strictly increasing in ¸ for any "xed j. Hence, one can always make ¸ large enough to ensure there is an interval [j0,j0] for which /(¸,j)'t(j). It is then evident that further increases in¸
will reduce j0, and increase j0. Finally, if ¸H
T'¸cT, we have EgHT!EgcT" :(j!1)a(¸H
Proof of Theorem 2.7. Assume for the moment that<
q`1is uniquely de"ned.
Substitute Eq. (11) into Eq. (10) to remove<
q(t):
one product generation and setting¸
q`1"0 (which is not optimal), we obtain
o<
uniquely determined. The uniqueness proof is then completed by backward induction to product generationq.
Next, note [from Eq. (A.2)] that¸H
n(aT~1). By backward induction to product generationq,
we can than conclude that¸H
q(t) and<q(t) are both increasing in each element of
the sequenceMME
n(aj)NTj/~1q, E(a)N.
Proof of Lemma 4.1. The conditional expectation is given by
E(¸H
where the second line was obtained with an integration by parts. As¸Hb
q (R) has
an upper bound at ¸b, d¸H
q/dEn(aq)'0 (Theorem 2.7), and H(En(aq)Daq) is
decreasing ina
q (Assumption 3.2), E(¸HqbDaq) is increasing inaq. Hence aq and
¸Hqb are positively correlated. We then have E(a
q¸Hqb)"cov(aq,¸Hqb)#
E(a
uninformative,His not a function ofa
qso thataqand¸Hqbare independent, and
cov(a
q,¸Hqb)"0.
Proof of Lemma 4.3. Di!erentiating Eq. (A.2) with respect to E
n(aq), one obtains
Using Eq. (A.2) to remove E
n(aq) from Eq. (A.4) yields, after some rearrangement,
d(¸H Eq. (A.5) is strictly positive and di!erentiable with respect to ¸H
q(t) for all
Eq. (A.6) is strictly positive. Hence, lim
LH
q(t)s0d(¸Hq(t)b)/dEn(aq)"0 i!b'1/2.
Note that Eq. (A.7) is di!erentiable in¸H
which is strictly increasing in¸H
q(t) for all¸Hq(t)/¸((b2lnj)/(b(1!b)j#(1!b)).
Proof of Proposition 4.4. Assume that E(¸H
q`1(t))*¸Tfor someq(¹!1. Then,
noting the dependence of¸H
q(t) on En(aq) and¸Hq`1(t), we can write
Inequality (i) is from Lemma 4.1; (ii) is from the convexity established in Lemma 4.3; (iii) is from Assumption 3.1; (iv) is the unique solution to Eq. (14) when E
n(aq)"E(a); and (v) is a restatement of Assumption 2.6.
The remainder of the proof is by induction. If E(¸H
q`1(t))5¸HT, we have
E(¸H
q(En(aq),¸Hq`1))5¸Hq(E(a),¸HT)
"¸H
T,
which follows from steps (i)}(iv) above, and the fact that¸H
Appendix B. Characterization of Example 3.3
Derivation of posterior density.Asa
qis exponentially distributed with parametervq,
the sumnkofnrealizations ofa
qhas a gamma distribution with parametersnandv.
The unconditional density fornkis
f
n(nk)"
P
=0 f
n(nkDv)f(v) dv
"
P
= 0b2vn`1(nk)n~1e~(b`nk)v C(n) dv.
Repeated integration by parts for integernyields
f
n(nk)"
b2(nk)n~1n(n#1)
(b#nk)n`2 (B.1)
and Bayes theorem yields the posterior density.
f
n(nkDv)" f
n(vDnk)f(v)
f
n(nk)
"vn`1e~(b`nk)v(b#nk)n`2 C(n#2) .
The posterior density ofa
qis then given by
f
n(aq)"
P
=0 ve~aqvf
n(vDnk) dv.
Combining the last two equations, and integrating by parts repeatedly for integern, yields Eq. (18).
Assumption 3.1. Decision makers in technologically lagging countries form expecta-tions as follows:
E
n(aq)"
P
=0 a
q(n#2)(b#nk)n`2
(a
q#b#nk)n`3
da q
"b#nk
The density ofkconditional on the unobserved true value ofa
The second line applies the method of transformations to Eq. (B.1); the third line combines Eqs. (17), (18) and (B.1). From Eq. (B.2), the posterior mean ofa
qand the
for all integer n. Hence, using Eq. (17), the unconditional expectation of E
n(aq) is
Thus Assumption 3.1 is satis"ed.
Assumption 3.2. The conditional distributionH(E
n(aq)Daq) is given by
Di!erentiating with respect toa
qand integrating by parts repeatedly yields
Finally,substituting Eq. (B.2) to remove E
n(aq) yields
dH(E
n(aq)Daq)
da q
"! (aq#b)2(nk)n
(a
q#b#bk)n`2
(0,
and Assumption 3.2 is satis"ed.
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